For what $a \in \Bbb R$ does $\neg(3a>21\implies a \leq 5)$ hold?

For what $a\in\mathbb{R}$ will the statement $\neg(3a>21\implies a\leq 5)$ hold?

My gut feeling says $a>7$, but I do not know how to formally write it down or prove it. Can someone help me?

$$\begin{align} \neg(3a>21\implies a\leq 5) & \;\equiv\; \neg(a \leq 5 \;\lor\; \neg(3a > 21)) \\ & \;\equiv\; \neg(a \leq 5) \;\land\; 3a > 21 \\ & \;\equiv\; a > 5 \;\land\; a > 7 \\ & \;\equiv\; a > 7 \end{align}$$

First note that $\neg(3a>21\implies a\le 5)$ is true exactly when $3a>21\implies a\le 5$ is false, so you really need to determine when $3a>21\implies a\le 5$ is false. I expect that you know that an implication $p\implies q$ is false if and only if $p$ is true and $q$ is false; in all other cases it’s true. Thus, we want to know for which values of $a$ the statement $3a>21$ is true and the statement $a\le 5$ is false. Now $3a>21$ is true exactly when then $a>\frac{21}3=7$; and whenever that’s the case, $a\le 5$ is certainly false. So for what values of $a$ is

$$\neg(3a>21\implies a\le 5)$$

true?